Relation between the convex roof extension and the least concave majorant extension
The strict relation between the coherence of convex roof and the coherence of assistance, that is, whether \(C_c(\rho )< C_a(\rho )\)  holds for all mixed quantum states is an interesting topic. The physical motivation is from the coherence distillation, which is to extract pure coherence from a mixed state by incoherent operations [1]}. All coherent states can be distilled by the coherence distillation process.
The assisted coherence distillation is then introduced to generate the maximal possible coherence with the help of another party's local measurements and classical communication [2]}. The relative entropy coherence of assistance in form of Eq. (REF ) with a specific function \(f\)  is proposed first there to quantify the one way coherence
distillation rate in the assisted coherence distillation. Generally we can get more coherence in the assisted coherence distillation. But a natural question is whether we can extract more coherence from all mixed states in the assisted coherence distillation. This question is factually equivalent to whether the coherence of assistance is strictly larger than the coherence of convex roof for all mixed quantum states. If it is true, all mixed quantum states are distillable in the assisted coherence distillation process. In order to answer this question, we consider a much more general case as follows.
